Large Deviations for Multi-valued Stochastic Differential Equations - - PowerPoint PPT Presentation

Large Deviations for Multi-valued Stochastic Differential Equations

Large Deviations for Multi-valued Stochastic Differential Equations

Joint work with Siyan Xu, Xicheng Zhang

We shall talk about the Freidlin-Wentzell Large Deviation Principle for the following multivalued stochastic differential equation

We shall talk about the Freidlin-Wentzell Large Deviation Principle for the following multivalued stochastic differential equation dXε(t) ∈ b(Xε(t)) dt + √εσ(Xε(t)) dW(t) − A(Xε(t)) dt, Xε(0) = x ∈ D(A), ε ∈ (0, 1].

We shall talk about the Freidlin-Wentzell Large Deviation Principle for the following multivalued stochastic differential equation dXε(t) ∈ b(Xε(t)) dt + √εσ(Xε(t)) dW(t) − A(Xε(t)) dt, Xε(0) = x ∈ D(A), ε ∈ (0, 1]. I.e., we seek a result of the following type:

We shall talk about the Freidlin-Wentzell Large Deviation Principle for the following multivalued stochastic differential equation dXε(t) ∈ b(Xε(t)) dt + √εσ(Xε(t)) dW(t) − A(Xε(t)) dt, Xε(0) = x ∈ D(A), ε ∈ (0, 1]. I.e., we seek a result of the following type: −I(F o)

• lim inf

ε↓0

ε log P(Xε ∈ F)

• lim sup

ε↓0

ε log P(Xε ∈ F)

• −I(F)

We shall talk about the Freidlin-Wentzell Large Deviation Principle for the following multivalued stochastic differential equation dXε(t) ∈ b(Xε(t)) dt + √εσ(Xε(t)) dW(t) − A(Xε(t)) dt, Xε(0) = x ∈ D(A), ε ∈ (0, 1]. I.e., we seek a result of the following type: −I(F o)

• lim inf

ε↓0

ε log P(Xε ∈ F)

• lim sup

ε↓0

ε log P(Xε ∈ F)

• −I(F)

for F ⊂ Cx([0, 1], D(A)),

We shall talk about the Freidlin-Wentzell Large Deviation Principle for the following multivalued stochastic differential equation dXε(t) ∈ b(Xε(t)) dt + √εσ(Xε(t)) dW(t) − A(Xε(t)) dt, Xε(0) = x ∈ D(A), ε ∈ (0, 1]. I.e., we seek a result of the following type: −I(F o)

• lim inf

ε↓0

ε log P(Xε ∈ F)

• lim sup

ε↓0

ε log P(Xε ∈ F)

• −I(F)

for F ⊂ Cx([0, 1], D(A)), where

We shall talk about the Freidlin-Wentzell Large Deviation Principle for the following multivalued stochastic differential equation dXε(t) ∈ b(Xε(t)) dt + √εσ(Xε(t)) dW(t) − A(Xε(t)) dt, Xε(0) = x ∈ D(A), ε ∈ (0, 1]. I.e., we seek a result of the following type: −I(F o)

• lim inf

ε↓0

ε log P(Xε ∈ F)

• lim sup

ε↓0

ε log P(Xε ∈ F)

• −I(F)

for F ⊂ Cx([0, 1], D(A)), where F o = the interior of F

We shall talk about the Freidlin-Wentzell Large Deviation Principle for the following multivalued stochastic differential equation dXε(t) ∈ b(Xε(t)) dt + √εσ(Xε(t)) dW(t) − A(Xε(t)) dt, Xε(0) = x ∈ D(A), ε ∈ (0, 1]. I.e., we seek a result of the following type: −I(F o)

• lim inf

ε↓0

ε log P(Xε ∈ F)

• lim sup

ε↓0

ε log P(Xε ∈ F)

• −I(F)

for F ⊂ Cx([0, 1], D(A)), where F o = the interior of F F = the closure of F

• I: a rate function on Cx([0, 1], D(A));

• I: a rate function on Cx([0, 1], D(A));

(I ∈ [0, ∞], {I c} is compact ∀c > 0).

• I: a rate function on Cx([0, 1], D(A));

(I ∈ [0, ∞], {I c} is compact ∀c > 0). and

• I: a rate function on Cx([0, 1], D(A));

(I ∈ [0, ∞], {I c} is compact ∀c > 0). and I(A) := inf

x∈A I(x)

• I: a rate function on Cx([0, 1], D(A));

(I ∈ [0, ∞], {I c} is compact ∀c > 0). and I(A) := inf

x∈A I(x)

Let us begin by explaining the notion of

• I: a rate function on Cx([0, 1], D(A));

(I ∈ [0, ∞], {I c} is compact ∀c > 0). and I(A) := inf

x∈A I(x)

Let us begin by explaining the notion of

MSDE: MULTIVALUED STOCHASTIC DIFFERENTIAL EQUATIONS

MSDE: MULTIVALUED STOCHASTIC DIFFERENTIAL EQUATIONS

dX(t) ∈ b(X(t))dt + σ(X(t))dw(t) − A(X(t))dt

MSDE: MULTIVALUED STOCHASTIC DIFFERENTIAL EQUATIONS

dX(t) ∈ b(X(t))dt + σ(X(t))dw(t) − A(X(t))dt Difference: An additional A

MSDE: MULTIVALUED STOCHASTIC DIFFERENTIAL EQUATIONS

dX(t) ∈ b(X(t))dt + σ(X(t))dw(t) − A(X(t))dt Difference: An additional A What is it?

MSDE: MULTIVALUED STOCHASTIC DIFFERENTIAL EQUATIONS

dX(t) ∈ b(X(t))dt + σ(X(t))dw(t) − A(X(t))dt Difference: An additional A What is it? A: a maximal monotone operator.

MSDE: MULTIVALUED STOCHASTIC DIFFERENTIAL EQUATIONS

dX(t) ∈ b(X(t))dt + σ(X(t))dw(t) − A(X(t))dt Difference: An additional A What is it? A: a maximal monotone operator. main features:

MSDE: MULTIVALUED STOCHASTIC DIFFERENTIAL EQUATIONS

dX(t) ∈ b(X(t))dt + σ(X(t))dw(t) − A(X(t))dt Difference: An additional A What is it? A: a maximal monotone operator. main features:

• A is multivalued: ∀x, A(x) is a set, not necessarily a single point

MSDE: MULTIVALUED STOCHASTIC DIFFERENTIAL EQUATIONS

dX(t) ∈ b(X(t))dt + σ(X(t))dw(t) − A(X(t))dt Difference: An additional A What is it? A: a maximal monotone operator. main features:

• A is multivalued: ∀x, A(x) is a set, not necessarily a single point
• A is monotone: ∀x, y, all u ∈ A(x), v ∈ A(y)

x − y, u − v 0

MSDE: MULTIVALUED STOCHASTIC DIFFERENTIAL EQUATIONS

dX(t) ∈ b(X(t))dt + σ(X(t))dw(t) − A(X(t))dt Difference: An additional A What is it? A: a maximal monotone operator. main features:

• A is multivalued: ∀x, A(x) is a set, not necessarily a single point
• A is monotone: ∀x, y, all u ∈ A(x), v ∈ A(y)

x − y, u − v 0

• A is maximal:

(x1, y1) ∈ Gr(A) ⇔ y1 − y2, x1 − x2 0, ∀(x2, y2) ∈ Gr(A).

Definition of Solution

Definition of Solution

Definition: A pair of continuous and Ft−adapted processes (X, K) is called a strong solution of

Definition of Solution

Definition: A pair of continuous and Ft−adapted processes (X, K) is called a strong solution of dX(t) ∈ b(X(t)) dt + σ(X(t)) dW(t) dt, X(0) = x ∈ D(A).

Definition of Solution

Definition: A pair of continuous and Ft−adapted processes (X, K) is called a strong solution of dX(t) ∈ b(X(t)) dt + σ(X(t)) dW(t) dt, X(0) = x ∈ D(A). if

(i) X0 = x0 and X(t) ∈ D(A) a.s., ∀t;

(i) X0 = x0 and X(t) ∈ D(A) a.s., ∀t; (ii) K = {K(t), Ft; t ∈ R+} is of finite variation and K(0) = 0 a.s.;

(i) X0 = x0 and X(t) ∈ D(A) a.s., ∀t; (ii) K = {K(t), Ft; t ∈ R+} is of finite variation and K(0) = 0 a.s.; (iii) dX(t) = b(X(t))dt + σ(X(t))dw(t) − dK(t), t ∈ R+, a.s.;

(i) X0 = x0 and X(t) ∈ D(A) a.s., ∀t; (ii) K = {K(t), Ft; t ∈ R+} is of finite variation and K(0) = 0 a.s.; (iii) dX(t) = b(X(t))dt + σ(X(t))dw(t) − dK(t), t ∈ R+, a.s.; (iv) for any continuous processes (α, β) satisfying (α(t), β(t)) ∈ Gr(A), ∀t ∈ R+,

(i) X0 = x0 and X(t) ∈ D(A) a.s., ∀t; (ii) K = {K(t), Ft; t ∈ R+} is of finite variation and K(0) = 0 a.s.; (iii) dX(t) = b(X(t))dt + σ(X(t))dw(t) − dK(t), t ∈ R+, a.s.; (iv) for any continuous processes (α, β) satisfying (α(t), β(t)) ∈ Gr(A), ∀t ∈ R+, the measure X(t) − α(t), dK(t) − β(t)dt

(i) X0 = x0 and X(t) ∈ D(A) a.s., ∀t; (ii) K = {K(t), Ft; t ∈ R+} is of finite variation and K(0) = 0 a.s.; (iii) dX(t) = b(X(t))dt + σ(X(t))dw(t) − dK(t), t ∈ R+, a.s.; (iv) for any continuous processes (α, β) satisfying (α(t), β(t)) ∈ Gr(A), ∀t ∈ R+, the measure X(t) − α(t), dK(t) − β(t)dt 0

(i) X0 = x0 and X(t) ∈ D(A) a.s., ∀t; (ii) K = {K(t), Ft; t ∈ R+} is of finite variation and K(0) = 0 a.s.; (iii) dX(t) = b(X(t))dt + σ(X(t))dw(t) − dK(t), t ∈ R+, a.s.; (iv) for any continuous processes (α, β) satisfying (α(t), β(t)) ∈ Gr(A), ∀t ∈ R+, the measure X(t) − α(t), dK(t) − β(t)dt 0 (Formally, this amounts to saying

(i) X0 = x0 and X(t) ∈ D(A) a.s., ∀t; (ii) K = {K(t), Ft; t ∈ R+} is of finite variation and K(0) = 0 a.s.; (iii) dX(t) = b(X(t))dt + σ(X(t))dw(t) − dK(t), t ∈ R+, a.s.; (iv) for any continuous processes (α, β) satisfying (α(t), β(t)) ∈ Gr(A), ∀t ∈ R+, the measure X(t) − α(t), dK(t) − β(t)dt 0 (Formally, this amounts to saying X(t) − α(t), K′(t)dt − β(t)dt 0

(i) X0 = x0 and X(t) ∈ D(A) a.s., ∀t; (ii) K = {K(t), Ft; t ∈ R+} is of finite variation and K(0) = 0 a.s.; (iii) dX(t) = b(X(t))dt + σ(X(t))dw(t) − dK(t), t ∈ R+, a.s.; (iv) for any continuous processes (α, β) satisfying (α(t), β(t)) ∈ Gr(A), ∀t ∈ R+, the measure X(t) − α(t), dK(t) − β(t)dt 0 (Formally, this amounts to saying X(t) − α(t), K′(t)dt − β(t)dt 0

• r

X(t) − α(t), K′(t) − β(t) 0 )

Existence and Uniqueness

Existence and Uniqueness

Theorem (C´ epa, 1995) If σ and b are Lipschitz, then dX(t) ∈ b(X(t)) dt + σ(X(t)) dW(t) − A(X(t)) dt, X(0) = x ∈ D(A), ε ∈ (0, 1].

Existence and Uniqueness

Theorem (C´ epa, 1995) If σ and b are Lipschitz, then dX(t) ∈ b(X(t)) dt + σ(X(t)) dW(t) − A(X(t)) dt, X(0) = x ∈ D(A), ε ∈ (0, 1]. admits a unique solution, (X, K).

Existence and Uniqueness

Theorem (C´ epa, 1995) If σ and b are Lipschitz, then dX(t) ∈ b(X(t)) dt + σ(X(t)) dW(t) − A(X(t)) dt, X(0) = x ∈ D(A), ε ∈ (0, 1]. admits a unique solution, (X, K). The unique solution of

Existence and Uniqueness

Theorem (C´ epa, 1995) If σ and b are Lipschitz, then dX(t) ∈ b(X(t)) dt + σ(X(t)) dW(t) − A(X(t)) dt, X(0) = x ∈ D(A), ε ∈ (0, 1]. admits a unique solution, (X, K). The unique solution of dXε(t) ∈ b(Xε(t)) dt + √εσ(Xε(t)) dW(t) − A(Xε(t)) dt, Xε(0) = x ∈ D(A), ε ∈ (0, 1].

Existence and Uniqueness

Theorem (C´ epa, 1995) If σ and b are Lipschitz, then dX(t) ∈ b(X(t)) dt + σ(X(t)) dW(t) − A(X(t)) dt, X(0) = x ∈ D(A), ε ∈ (0, 1]. admits a unique solution, (X, K). The unique solution of dXε(t) ∈ b(Xε(t)) dt + √εσ(Xε(t)) dW(t) − A(Xε(t)) dt, Xε(0) = x ∈ D(A), ε ∈ (0, 1]. will be denoted by

Existence and Uniqueness

Theorem (C´ epa, 1995) If σ and b are Lipschitz, then dX(t) ∈ b(X(t)) dt + σ(X(t)) dW(t) − A(X(t)) dt, X(0) = x ∈ D(A), ε ∈ (0, 1]. admits a unique solution, (X, K). The unique solution of dXε(t) ∈ b(Xε(t)) dt + √εσ(Xε(t)) dW(t) − A(Xε(t)) dt, Xε(0) = x ∈ D(A), ε ∈ (0, 1]. will be denoted by (Xε, Kε)

normally reflected SDE as a special case of MSDE

normally reflected SDE as a special case of MSDE

Suppose

normally reflected SDE as a special case of MSDE

Suppose O: a closed convex subset of Rm,

normally reflected SDE as a special case of MSDE

Suppose O: a closed convex subset of Rm, IO: the indicator function of O,

normally reflected SDE as a special case of MSDE

Suppose O: a closed convex subset of Rm, IO: the indicator function of O, i.e, IO(x) =

normally reflected SDE as a special case of MSDE

Suppose O: a closed convex subset of Rm, IO: the indicator function of O, i.e, IO(x) = 0, if x ∈ O,

normally reflected SDE as a special case of MSDE

Suppose O: a closed convex subset of Rm, IO: the indicator function of O, i.e, IO(x) = 0, if x ∈ O, + ∞, if x / ∈ O.

normally reflected SDE as a special case of MSDE

Suppose O: a closed convex subset of Rm, IO: the indicator function of O, i.e, IO(x) = 0, if x ∈ O, + ∞, if x / ∈ O. The subdifferential of IO is given by

normally reflected SDE as a special case of MSDE

Suppose O: a closed convex subset of Rm, IO: the indicator function of O, i.e, IO(x) = 0, if x ∈ O, + ∞, if x / ∈ O. The subdifferential of IO is given by ∂IO(x) = {y ∈ Rm : y, x − zRm 0, ∀z ∈ O}

normally reflected SDE as a special case of MSDE

Suppose O: a closed convex subset of Rm, IO: the indicator function of O, i.e, IO(x) = 0, if x ∈ O, + ∞, if x / ∈ O. The subdifferential of IO is given by ∂IO(x) = {y ∈ Rm : y, x − zRm 0, ∀z ∈ O} =   

normally reflected SDE as a special case of MSDE

Suppose O: a closed convex subset of Rm, IO: the indicator function of O, i.e, IO(x) = 0, if x ∈ O, + ∞, if x / ∈ O. The subdifferential of IO is given by ∂IO(x) = {y ∈ Rm : y, x − zRm 0, ∀z ∈ O} =    ∅, if x / ∈ O,

normally reflected SDE as a special case of MSDE

Suppose O: a closed convex subset of Rm, IO: the indicator function of O, i.e, IO(x) = 0, if x ∈ O, + ∞, if x / ∈ O. The subdifferential of IO is given by ∂IO(x) = {y ∈ Rm : y, x − zRm 0, ∀z ∈ O} =    ∅, if x / ∈ O, {0}, if x ∈ Int(O),

normally reflected SDE as a special case of MSDE

Suppose O: a closed convex subset of Rm, IO: the indicator function of O, i.e, IO(x) = 0, if x ∈ O, + ∞, if x / ∈ O. The subdifferential of IO is given by ∂IO(x) = {y ∈ Rm : y, x − zRm 0, ∀z ∈ O} =    ∅, if x / ∈ O, {0}, if x ∈ Int(O), Λx, if x ∈ ∂O,

normally reflected SDE as a special case of MSDE

Suppose O: a closed convex subset of Rm, IO: the indicator function of O, i.e, IO(x) = 0, if x ∈ O, + ∞, if x / ∈ O. The subdifferential of IO is given by ∂IO(x) = {y ∈ Rm : y, x − zRm 0, ∀z ∈ O} =    ∅, if x / ∈ O, {0}, if x ∈ Int(O), Λx, if x ∈ ∂O, where

normally reflected SDE as a special case of MSDE

Suppose O: a closed convex subset of Rm, IO: the indicator function of O, i.e, IO(x) = 0, if x ∈ O, + ∞, if x / ∈ O. The subdifferential of IO is given by ∂IO(x) = {y ∈ Rm : y, x − zRm 0, ∀z ∈ O} =    ∅, if x / ∈ O, {0}, if x ∈ Int(O), Λx, if x ∈ ∂O, where Int(O) is the interior of O

normally reflected SDE as a special case of MSDE

Suppose O: a closed convex subset of Rm, IO: the indicator function of O, i.e, IO(x) = 0, if x ∈ O, + ∞, if x / ∈ O. The subdifferential of IO is given by ∂IO(x) = {y ∈ Rm : y, x − zRm 0, ∀z ∈ O} =    ∅, if x / ∈ O, {0}, if x ∈ Int(O), Λx, if x ∈ ∂O, where Int(O) is the interior of O Λx is the exterior normal cone at x.

Then ∂IO is a multivalued maximal monotone operator.

Then ∂IO is a multivalued maximal monotone operator. In this case, an MSDE is an SDE normally reflected at the boundary of O.

Then ∂IO is a multivalued maximal monotone operator. In this case, an MSDE is an SDE normally reflected at the boundary of O. More generally,

Then ∂IO is a multivalued maximal monotone operator. In this case, an MSDE is an SDE normally reflected at the boundary of O. More generally, the subdifferential of any convex and lower semi-continuous function is a maximal monotone operator.

Then ∂IO is a multivalued maximal monotone operator. In this case, an MSDE is an SDE normally reflected at the boundary of O. More generally, the subdifferential of any convex and lower semi-continuous function is a maximal monotone operator. ∂ϕ(x) := {y ∈ Rm : y, z − xRm ϕ(z) − ϕ(x), ∀z ∈ Rm}

Then ∂IO is a multivalued maximal monotone operator. In this case, an MSDE is an SDE normally reflected at the boundary of O. More generally, the subdifferential of any convex and lower semi-continuous function is a maximal monotone operator. ∂ϕ(x) := {y ∈ Rm : y, z − xRm ϕ(z) − ϕ(x), ∀z ∈ Rm}

Towards a Freidlin-Wentzell theory

Towards a Freidlin-Wentzell theory

We now look at the problem of small perturbation:

Towards a Freidlin-Wentzell theory

We now look at the problem of small perturbation: dXε(t) ∈ b(Xε(t)) dt + √εσ(Xε(t)) dW(t) − A(Xε(t)) dt, Xε(0) = x ∈ D(A), ε ∈ (0, 1].

Classical theory

Classical theory

Recall the classical case:

Classical theory

Recall the classical case: A ≡ 0

Classical theory

Recall the classical case: A ≡ 0 dXε(t) = b(Xε(t)) dt + √εσ(Xε(t)) dW(t), Xε(0) = x, ε ∈ (0, 1].

Classical theory

Recall the classical case: A ≡ 0 dXε(t) = b(Xε(t)) dt + √εσ(Xε(t)) dW(t), Xε(0) = x, ε ∈ (0, 1]. The theory begins in the seminal paper

Classical theory

Recall the classical case: A ≡ 0 dXε(t) = b(Xε(t)) dt + √εσ(Xε(t)) dW(t), Xε(0) = x, ε ∈ (0, 1]. The theory begins in the seminal paper Wentzell-Freidlin: On small random perturbation of dynamical system, 1969

Classical theory

Recall the classical case: A ≡ 0 dXε(t) = b(Xε(t)) dt + √εσ(Xε(t)) dW(t), Xε(0) = x, ε ∈ (0, 1]. The theory begins in the seminal paper Wentzell-Freidlin: On small random perturbation of dynamical system, 1969 There are two standard and well known approaches to this classical problem.

• 1. Use of Euler Scheme

• 1. Use of Euler Scheme

D.W.Stroock: An introduction to the theory of large deviation, 1984

• 1. Use of Euler Scheme

D.W.Stroock: An introduction to the theory of large deviation, 1984 You look at the Euler Scheme

• 1. Use of Euler Scheme

D.W.Stroock: An introduction to the theory of large deviation, 1984 You look at the Euler Scheme Xε

n(t) = x +

t b(Xε

n(n−1[sn])) ds + √ε

t σ(Xε

n(n−1[sn])) dW(s),

• 1. Use of Euler Scheme

D.W.Stroock: An introduction to the theory of large deviation, 1984 You look at the Euler Scheme Xε

n(t) = x +

t b(Xε

n(n−1[sn])) ds + √ε

t σ(Xε

n(n−1[sn])) dW(s),

Then you have to estimate the quantity

• 1. Use of Euler Scheme

D.W.Stroock: An introduction to the theory of large deviation, 1984 You look at the Euler Scheme Xε

n(t) = x +

t b(Xε

n(n−1[sn])) ds + √ε

t σ(Xε

n(n−1[sn])) dW(s),

Then you have to estimate the quantity P( max

0t1 |Xε(t, x) − Xε n| δ)

• 1. Use of Euler Scheme

D.W.Stroock: An introduction to the theory of large deviation, 1984 You look at the Euler Scheme Xε

n(t) = x +

t b(Xε

n(n−1[sn])) ds + √ε

t σ(Xε

n(n−1[sn])) dW(s),

Then you have to estimate the quantity P( max

0t1 |Xε(t, x) − Xε n| δ)

If you have for any δ > 0

• 1. Use of Euler Scheme

D.W.Stroock: An introduction to the theory of large deviation, 1984 You look at the Euler Scheme Xε

n(t) = x +

t b(Xε

n(n−1[sn])) ds + √ε

t σ(Xε

n(n−1[sn])) dW(s),

Then you have to estimate the quantity P( max

0t1 |Xε(t, x) − Xε n| δ)

If you have for any δ > 0 lim

n→∞ lim sup ε↓0

ε log P( max

0t1 |Xε(t, x) − Xε n| δ) = −∞,

• 1. Use of Euler Scheme

D.W.Stroock: An introduction to the theory of large deviation, 1984 You look at the Euler Scheme Xε

n(t) = x +

t b(Xε

n(n−1[sn])) ds + √ε

t σ(Xε

n(n−1[sn])) dW(s),

Then you have to estimate the quantity P( max

0t1 |Xε(t, x) − Xε n| δ)

If you have for any δ > 0 lim

n→∞ lim sup ε↓0

ε log P( max

0t1 |Xε(t, x) − Xε n| δ) = −∞,

then you are done.

Unfortunately, this approach does NOT apply to MSDE case

Unfortunately, this approach does NOT apply to MSDE case simply because the Euler scheme cannot be defined for MSDE

Unfortunately, this approach does NOT apply to MSDE case simply because the Euler scheme cannot be defined for MSDE since either A(Xn(t)) is not defined or may be a set.

• 2. Use of Freidlin-Wentzell estimate

• 2. Use of Freidlin-Wentzell estimate

(R. Azencott: Grandes deviations et applications, 1980

• 2. Use of Freidlin-Wentzell estimate

(R. Azencott: Grandes deviations et applications, 1980

• P. Priouret: Remarques sur les petite perturbations de syst`

emes dynamiques, 1982)

• 2. Use of Freidlin-Wentzell estimate

(R. Azencott: Grandes deviations et applications, 1980

• P. Priouret: Remarques sur les petite perturbations de syst`

emes dynamiques, 1982) Consider an ODE

• 2. Use of Freidlin-Wentzell estimate

(R. Azencott: Grandes deviations et applications, 1980

• P. Priouret: Remarques sur les petite perturbations de syst`

emes dynamiques, 1982) Consider an ODE dg(t) = b(g(t)) dt + σ(g(t))f′(t) dt, g(0) = x.

• 2. Use of Freidlin-Wentzell estimate

(R. Azencott: Grandes deviations et applications, 1980

• P. Priouret: Remarques sur les petite perturbations de syst`

emes dynamiques, 1982) Consider an ODE dg(t) = b(g(t)) dt + σ(g(t))f′(t) dt, g(0) = x. where f satisfies

• 2. Use of Freidlin-Wentzell estimate

(R. Azencott: Grandes deviations et applications, 1980

• P. Priouret: Remarques sur les petite perturbations de syst`

emes dynamiques, 1982) Consider an ODE dg(t) = b(g(t)) dt + σ(g(t))f′(t) dt, g(0) = x. where f satisfies 1 f′(t)2dt < ∞.

If you can prove

If you can prove ∀R > 0,

If you can prove ∀R > 0, ρ > 0,

If you can prove ∀R > 0, ρ > 0, ∃ε0 > 0,

If you can prove ∀R > 0, ρ > 0, ∃ε0 > 0, α > 0,

If you can prove ∀R > 0, ρ > 0, ∃ε0 > 0, α > 0, r > 0

If you can prove ∀R > 0, ρ > 0, ∃ε0 > 0, α > 0, r > 0 such that

If you can prove ∀R > 0, ρ > 0, ∃ε0 > 0, α > 0, r > 0 such that ε log P(Xε − g > ρ, ε

1 2 W − f < α) −R,

∀ε ε0

If you can prove ∀R > 0, ρ > 0, ∃ε0 > 0, α > 0, r > 0 such that ε log P(Xε − g > ρ, ε

1 2 W − f < α) −R,

∀ε ε0 then you have the large deviation principle.

But this approach does NOT apply neither

But this approach does NOT apply neither since to obtain the Freidlin-Wentzell estimate directly

But this approach does NOT apply neither since to obtain the Freidlin-Wentzell estimate directly

• ne has to suppose b and σ are Lipschitz

But this approach does NOT apply neither since to obtain the Freidlin-Wentzell estimate directly

• ne has to suppose b and σ are Lipschitz

(Priouret had to.).

• 3. Use of weak convergence

• 3. Use of weak convergence

This approach was developed a few years ago by

• 3. Use of weak convergence

This approach was developed a few years ago by

• P. Dupuis and R.S. Ellis: A weak convergence approach to the

theory of large deviatins, 1997

• 3. Use of weak convergence

This approach was developed a few years ago by

• P. Dupuis and R.S. Ellis: A weak convergence approach to the

theory of large deviatins, 1997 and was tested in several recent work to treat Wentzell-Freidlin large deviation principle for SDEs with irregular coefficients. It has proved to be highly efficient.

• 3. Use of weak convergence

This approach was developed a few years ago by

• P. Dupuis and R.S. Ellis: A weak convergence approach to the

theory of large deviatins, 1997 and was tested in several recent work to treat Wentzell-Freidlin large deviation principle for SDEs with irregular coefficients. It has proved to be highly efficient. The hard core of this approach is the following observation

• Xn: a sequence of r.v., valued in a complete metric space X ;

• Xn: a sequence of r.v., valued in a complete metric space X ;
• I: a rate function on X ;

• Xn: a sequence of r.v., valued in a complete metric space X ;
• I: a rate function on X ;

{Xn} is said to satisfy the LDP with rate function I if

• Xn: a sequence of r.v., valued in a complete metric space X ;
• I: a rate function on X ;

{Xn} is said to satisfy the LDP with rate function I if lim sup

n→∞

1 n log P(Xn ∈ F) −I(F), F closed

• Xn: a sequence of r.v., valued in a complete metric space X ;
• I: a rate function on X ;

{Xn} is said to satisfy the LDP with rate function I if lim sup

n→∞

1 n log P(Xn ∈ F) −I(F), F closed lim inf

n→∞

1 n log P(Xn ∈ G) −I(G), F open {Xn} is said to satisfy the Laplace Principle with rate function I if

• Xn: a sequence of r.v., valued in a complete metric space X ;
• I: a rate function on X ;

{Xn} is said to satisfy the LDP with rate function I if lim sup

n→∞

1 n log P(Xn ∈ F) −I(F), F closed lim inf

n→∞

1 n log P(Xn ∈ G) −I(G), F open {Xn} is said to satisfy the Laplace Principle with rate function I if lim

n→∞

1 n log E{exp[−nh(Xn)]} = − inf

x∈X {h(x) + I(x)}.

Theorem: LP⇐ ⇒ LDP

Theorem: LP⇐ ⇒ LDP (⇐ =: Varadhan, 1966; = ⇒: Dupuis-Ellis, 1997)

Theorem: LP⇐ ⇒ LDP (⇐ =: Varadhan, 1966; = ⇒: Dupuis-Ellis, 1997) Now let us return to our MSDE.

Theorem: LP⇐ ⇒ LDP (⇐ =: Varadhan, 1966; = ⇒: Dupuis-Ellis, 1997) Now let us return to our MSDE. Suppose D ⊂ Rd is an open set. Consider the following reflected SDE

Theorem: LP⇐ ⇒ LDP (⇐ =: Varadhan, 1966; = ⇒: Dupuis-Ellis, 1997) Now let us return to our MSDE. Suppose D ⊂ Rd is an open set. Consider the following reflected SDE dXε(t) ∈ b(Xε(t))dt+σ(Xε(t))◦dw(t)−v(Xε(t))da(t), X(0) = x

Theorem: LP⇐ ⇒ LDP (⇐ =: Varadhan, 1966; = ⇒: Dupuis-Ellis, 1997) Now let us return to our MSDE. Suppose D ⊂ Rd is an open set. Consider the following reflected SDE dXε(t) ∈ b(Xε(t))dt+σ(Xε(t))◦dw(t)−v(Xε(t))da(t), X(0) = x where a increases only when Xε is on the boundary of D, v(x) is in the outward normal cone of D at x ∈ ∂D.

Theorem: LP⇐ ⇒ LDP (⇐ =: Varadhan, 1966; = ⇒: Dupuis-Ellis, 1997) Now let us return to our MSDE. Suppose D ⊂ Rd is an open set. Consider the following reflected SDE dXε(t) ∈ b(Xε(t))dt+σ(Xε(t))◦dw(t)−v(Xε(t))da(t), X(0) = x where a increases only when Xε is on the boundary of D, v(x) is in the outward normal cone of D at x ∈ ∂D. For the result of the following type

Theorem: LP⇐ ⇒ LDP (⇐ =: Varadhan, 1966; = ⇒: Dupuis-Ellis, 1997) Now let us return to our MSDE. Suppose D ⊂ Rd is an open set. Consider the following reflected SDE dXε(t) ∈ b(Xε(t))dt+σ(Xε(t))◦dw(t)−v(Xε(t))da(t), X(0) = x where a increases only when Xε is on the boundary of D, v(x) is in the outward normal cone of D at x ∈ ∂D. For the result of the following type −I(F o)

• lim inf

ε↓0

ε log P(Xε ∈ F)

• lim sup

ε↓0

ε log P(Xε ∈ F)

• −I(F)

Theorem: LP⇐ ⇒ LDP (⇐ =: Varadhan, 1966; = ⇒: Dupuis-Ellis, 1997) Now let us return to our MSDE. Suppose D ⊂ Rd is an open set. Consider the following reflected SDE dXε(t) ∈ b(Xε(t))dt+σ(Xε(t))◦dw(t)−v(Xε(t))da(t), X(0) = x where a increases only when Xε is on the boundary of D, v(x) is in the outward normal cone of D at x ∈ ∂D. For the result of the following type −I(F o)

• lim inf

ε↓0

ε log P(Xε ∈ F)

• lim sup

ε↓0

ε log P(Xε ∈ F)

• −I(F)

for F ⊂ Cx([0, 1], D(A)),

Theorem: LP⇐ ⇒ LDP (⇐ =: Varadhan, 1966; = ⇒: Dupuis-Ellis, 1997) Now let us return to our MSDE. Suppose D ⊂ Rd is an open set. Consider the following reflected SDE dXε(t) ∈ b(Xε(t))dt+σ(Xε(t))◦dw(t)−v(Xε(t))da(t), X(0) = x where a increases only when Xε is on the boundary of D, v(x) is in the outward normal cone of D at x ∈ ∂D. For the result of the following type −I(F o)

• lim inf

ε↓0

ε log P(Xε ∈ F)

• lim sup

ε↓0

ε log P(Xε ∈ F)

• −I(F)

for F ⊂ Cx([0, 1], D(A)), the story is

(i) by Anderson and Orey in 1976 when D is (in addition to convex) smooth and σσ∗ > 0;

(i) by Anderson and Orey in 1976 when D is (in addition to convex) smooth and σσ∗ > 0; (ii) by Doss and Priouret when σ may be degenerated;

(i) by Anderson and Orey in 1976 when D is (in addition to convex) smooth and σσ∗ > 0; (ii) by Doss and Priouret when σ may be degenerated; (iii) by Dupuis for general convex sets D

(i) by Anderson and Orey in 1976 when D is (in addition to convex) smooth and σσ∗ > 0; (ii) by Doss and Priouret when σ may be degenerated; (iii) by Dupuis for general convex sets D

Return to our original problem.

Return to our original problem. If we manage to prove a Laplace principle, then we are done.

Return to our original problem. If we manage to prove a Laplace principle, then we are done. But how to obtain the Laplace principle?

Return to our original problem. If we manage to prove a Laplace principle, then we are done. But how to obtain the Laplace principle? Budhiraja and Dupuis pointed out the way for us.

Things to do

Things to do

Set B := C([0, 1], Rd)

Things to do

Set B := C([0, 1], Rd) H := {h ∈ B such that h is a. c. and h2

H :=

1 |˙ h(s)|2ds < ∞}

Things to do

Set B := C([0, 1], Rd) H := {h ∈ B such that h is a. c. and h2

H :=

1 |˙ h(s)|2ds < ∞} µ the standard Wiener measure on B

Things to do

Set B := C([0, 1], Rd) H := {h ∈ B such that h is a. c. and h2

H :=

1 |˙ h(s)|2ds < ∞} µ the standard Wiener measure on B Then (B, H, µ) is a classical Wiener space.

Things to do

Set B := C([0, 1], Rd) H := {h ∈ B such that h is a. c. and h2

H :=

1 |˙ h(s)|2ds < ∞} µ the standard Wiener measure on B Then (B, H, µ) is a classical Wiener space. ∀N > 0, let

Things to do

Set B := C([0, 1], Rd) H := {h ∈ B such that h is a. c. and h2

H :=

1 |˙ h(s)|2ds < ∞} µ the standard Wiener measure on B Then (B, H, µ) is a classical Wiener space. ∀N > 0, let DN := {h ∈ H : hH N}

Things to do

Set B := C([0, 1], Rd) H := {h ∈ B such that h is a. c. and h2

H :=

1 |˙ h(s)|2ds < ∞} µ the standard Wiener measure on B Then (B, H, µ) is a classical Wiener space. ∀N > 0, let DN := {h ∈ H : hH N} Then DN with weak topology is a metrizable compact Polish space.

Things to do

Set B := C([0, 1], Rd) H := {h ∈ B such that h is a. c. and h2

H :=

1 |˙ h(s)|2ds < ∞} µ the standard Wiener measure on B Then (B, H, µ) is a classical Wiener space. ∀N > 0, let DN := {h ∈ H : hH N} Then DN with weak topology is a metrizable compact Polish space. AN := {h : h is an adapted process , h(·) ∈ DN a.s.}

The way pointed out by Budhiraja and Duuis

The way pointed out by Budhiraja and Duuis

Let Yn : B → X , n = 1, 2, · · · .

The way pointed out by Budhiraja and Duuis

Let Yn : B → X , n = 1, 2, · · · . If there exists Y0 : H → X such that

The way pointed out by Budhiraja and Duuis

Let Yn : B → X , n = 1, 2, · · · . If there exists Y0 : H → X such that A For any N > 0, if a family {hn} ⊂ AN (as random variables in DN) converges in distribution to h ∈ AN, then for some subsequence nk, Ynk

• w +

hnk(w)

• n−1

k

• converges in distribution to

Y0(h) in X .

The way pointed out by Budhiraja and Duuis

Let Yn : B → X , n = 1, 2, · · · . If there exists Y0 : H → X such that A For any N > 0, if a family {hn} ⊂ AN (as random variables in DN) converges in distribution to h ∈ AN, then for some subsequence nk, Ynk

• w +

hnk(w)

• n−1

k

• converges in distribution to

Y0(h) in X . B For any N > 0, if {hn, n ∈ N} ⊂ AN converges weakly to h ∈ H, then for some subsequence hnk, Y0(hnk) converges weakly to Y0(h) in X .

Theorem: Under (A) and (B), the LP holds: lim

n→∞ log E{exp[−nf(Yn)]} = − inf x∈X {g(x) + I(x)}

Verification of (A) and (B)

Verification of (A) and (B)

We replace n by ε−1 and we take

Verification of (A) and (B)

We replace n by ε−1 and we take Y0 = X

Verification of (A) and (B)

We replace n by ε−1 and we take Y0 = X Yε := Xε,hε.

Verification of (A) and (B)

We replace n by ε−1 and we take Y0 = X Yε := Xε,hε. Verification of (A).

Verification of (A) and (B)

We replace n by ε−1 and we take Y0 = X Yε := Xε,hε. Verification of (A). Now Yε satisfies:

Verification of (A) and (B)

We replace n by ε−1 and we take Y0 = X Yε := Xε,hε. Verification of (A). Now Yε satisfies: Yε(t) = x + t b(Yε(s)) ds + t σ(Yε(s))˙ hε(s) ds +√ε t σ(Yε(s)) dW(s) − Kε(t)

Verification of (A) and (B)

We replace n by ε−1 and we take Y0 = X Yε := Xε,hε. Verification of (A). Now Yε satisfies: Yε(t) = x + t b(Yε(s)) ds + t σ(Yε(s))˙ hε(s) ds +√ε t σ(Yε(s)) dW(s) − Kε(t) Y0(t) = x + t b(Y0(s)) ds + t σ(Y0(s))˙ h(s) ds − K0(t).

Note (A) is equivalent to the apparently weaker condition

Note (A) is equivalent to the apparently weaker condition A’ For any N > 0, if a family {hn} ⊂ AN (as random variables in DN) converges almost surely to h ∈ AN, in H, then for some subsequence nk, Ynk

• w +

hnk(w)

• n−1

k

• converges in distribution to

Y0(h) in X .

Note (A) is equivalent to the apparently weaker condition A’ For any N > 0, if a family {hn} ⊂ AN (as random variables in DN) converges almost surely to h ∈ AN, in H, then for some subsequence nk, Ynk

• w +

hnk(w)

• n−1

k

• converges in distribution to

Y0(h) in X . This equivalence can be easily achieved by using Skorohod representation.

Hence, to prove (A), it suffices to prove that Yε − → Y0, weakly

Hence, to prove (A), it suffices to prove that Yε − → Y0, weakly if hε − → h0 almost surely in H.

Hence, to prove (A), it suffices to prove that Yε − → Y0, weakly if hε − → h0 almost surely in H. This can be done by Itˆ

• calculus.

Hence, to prove (A), it suffices to prove that Yε − → Y0, weakly if hε − → h0 almost surely in H. This can be done by Itˆ

• calculus.

Similarly, B can be verified.